7.3.8 problem 8

Internal problem ID [2325]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.4. Page 24
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 05:26:48 AM
CAS classification : [_separable]

\begin{align*} \sqrt {t^{2}+1}\, y^{\prime }&=\frac {t y^{3}}{\sqrt {t^{2}+1}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 16

ode:=(t^2+1)^(1/2)*diff(y(t),t) = t*y(t)^3/(t^2+1)^(1/2); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {1}{\sqrt {1-\ln \left (t^{2}+1\right )}} \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 19

ode=(t^2+1)^(1/2)*D[y[t],t] == t*y[t]^3/(t^2+1)^(1/2); 
ic=y[0]==1; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{\sqrt {1-\log \left (t^2+1\right )}} \end{align*}

✓ Sympy. Time used: 0.428 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*y(t)**3/sqrt(t**2 + 1) + sqrt(t**2 + 1)*Derivative(y(t), t),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \sqrt {- \frac {1}{\log {\left (t^{2} + 1 \right )} - 1}} \]